Another proof of $p_c=\frac{\sqrt{q}}{1+\sqrt{q}}$ on ${\mathbb{Z}}^2$ with $q\in [1,4]$ for random-cluster model

Abstract

In this paper we give another proof of $p_c=\frac{\sqrt{q}}{1+\sqrt{q}}$ on ${\mathbb{Z}}^2$ with $q\in [1,4]$, based on the method of parafermionic observables.

29 pages, 6 figures